Holt's first game in charge of Falkirk looked like it would be memorable for the right reasons as his team raced into a 3-0 lead after half-an-hour.
But the Premier League side rallied after the break and completed an unlikely comeback five minutes before the end of extra-time at Hampden.
"It's a proud day and I told them they should be proud," Holt said.
"I told them to give everything and to come off the pitch on their hands and knees and if they do that, the result will take care of itself.
"They've given everything but we couldn't quite end it.
"We knew they were going to come out hammer and tongs and they did that. If we could have just held on for a little bit longer it might have been different."